Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear operator. Many important conjectures involving polynomial rings, such as Serre's problem, have influenced the study of other rings, and have influenced even the definition of other rings, such as group rings and rings of formal power series.
A closely related notion is that of the ring of polynomial functions on a vector space.
The polynomial ring, K[X], in X over a field K is defined as the set of expressions, called polynomials in X, of the form
where p0, p1,…, pm, the coefficients of p, are elements of K, and X, X2, are symbols, which are considered as "powers of X", and, by convention, follow the usual rules of exponentiation: X0 = 1, X1 = X, and
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